Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Affinity laws — where it appears

The statements that a machine's flow, head and power scale as fixed powers of its speed and diameter, which is what geometric similarity gives. They hold exactly for a machine at unchanged dimensionless operating point, and they are how one measured characteristic serves a whole family.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

1.139 asks for a Francis. The specific-speed axis, with the four machine types on it and one duty marked: 3 m³/s at 60 m, on a shaft turning at 750 rev/min. The number is 1.1387, and the choice of runner follows from it before any blade has been drawn. What the number contains is a ratio of flow to head; what it does not contain is any size at all, which is why one axis serves a garden pump and a gigawatt turbine.

One number picks the machine

A flow rate, a head and a shaft speed contain exactly one dimensionless combination with no size in it. That combination decides whether a duty wants an impulse wheel, a Francis runner or a propeller — before anything has been drawn, sized, or costed.

applied · Specific speed Two ways to move a duty, along one axis, in opposite directions. A duty at a specific speed of 0.01 and what splitting it does. Dividing the head between stages in series multiplies each stage's specific speed by the number of stages to the three-quarter power, because the group carries (gH)^(−3/4); dividing the flow between units in parallel divides it by the square root of the number of units, because the group carries √Q. Both exponents are read off the group rather than remembered, and they are why the two operations are not interchangeable: three stages buy a factor of 2.28 and three units cost a factor of 0.58. The bands are drawn in the colour this site reserves for a borrowed claim, because where a Francis runner stops is practice rather than a result.

The duty that had no machine

Some duties have a specific speed outside every band, and no runner will do them at any size because the number contains no size. That is true and it is not the end. The same group says how to split the duty until it fits, and the two ways of splitting move it along the same axis in opposite directions, by exponents read straight off it.

applied · Specific speed A pump slowed into a system with a static lift leaves its own specific speed. The specific speed of the operating point, as a share of its value at the best point, against the fraction of the design flow delivered, for a pump drawn for 0.1 m³/s against 40 m at 1450 rpm, specific speed 0.545 at its best point. Under speed control into a system whose static lift is 0 per cent, 30 per cent, 60 per cent, 90 per cent of the design head (lines), and under a throttle at design speed into the 60 per cent system (dashed). With no static lift the speed-controlled pump stays exactly at its best point and its specific speed never moves. At half the design flow it has fallen to 1.000, 0.800, 0.708, 0.652 of the design value as the static share rises, and to 0.598 under the throttle.

The specific speed a pump spends its life at

A pump is chosen by its specific speed at its best point and then run somewhere else. Written in the pump's own coefficients the number is √φ/ψ^¾, a position along its characteristic, and a variable-speed drive keeps it there only when the system it pumps into has no static lift. Every metre of lift moves a slowed pump along its own curve, towards shut-off, and a throttle moves it further.

applied · Specific speed One fast pump or two slow ones. Pump efficiency against the flow asked for, for two pumps into a system with 60 per cent static lift: one pump under speed control, up to the flow at which it reaches full speed; both under speed control; and the best number running with the pumps held at full speed and a valve taking up the difference. Below 0.601 of the design flow one pump is better than two, and above it two are better — before the single pump has run out of speed at 0.65. The throttle is far worse at every flow.

One fast pump or two slow ones

A pumping station with several identical pumps in parallel has a control that a single pump does not: how many of them are running. With no static lift the answer is all of them, always, slowed together. With lift it is a number that falls with the flow and is almost never whole — the flow times the number of pumps, over the square root of the head the system asks. Staging rounds it, and the right moment to round up comes before the running pumps have run out of speed.

applied · Specific speed Below a specific speed of a half, a pump's losses are its discs and its narrow passages. Each loss as a share of the shaft power against the specific speed, for pumps of 40 m head at 1450 rpm, so that every one has the same impeller diameter and only the flow changes. At Ωₛ = 1.0 the fixed share is most of the loss and the efficiency is 0.902; at 0.3, disc friction is 7.65% and passage friction 5.32%, and the efficiency 0.777; at 0.15 the two together take 37.9%.

Two small pumps carry two big discs

Split a pumping duty between two pumps of the same head and speed and each one keeps the big pump's impeller diameter, because the head fixes the tip speed. So the pair spins two full-size discs in their casings for the flow one disc used to carry, and the efficiency penalty that a station comparison assumed to be two points is four. That moves the lift at which splitting saves energy from 44 per cent to 65, and below a specific speed of about a half it never does.

applied · Specific speed

Named alongside it

The objects these essays reach for when they reach for this one.

SimilaritySpecific speedOptimisationCorrelationDimensionlessEfficiencyModel limitScalingDimensional analysisPipe flowPump characteristicRank

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